Using Toroidal Coordinates to Determine the Gravitational Potential

NOTE: An earlier version of this chapter has been shifted to our "Ramblings" Appendix.

Here we build upon our accompanying review of the types of numerical techniques that various astrophysics research groups have developed to solve for the Newtonian gravitational potential, <math>~\Phi(\vec{x})</math>, given a specified, three-dimensional mass distribution, <math>~\rho(\vec{x})</math>. Our focus is on the use of toroidal coordinates to solve the integral formulation of the Poisson equation, namely,

<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>

For the most part, we will adopt the notation used by C.-Y. Wong (1973, Annals of Physics, 77, 279); in an accompanying discussion, we review additional results from this insightful 1973 paper, as well as a paper of his that was published the following year in The Astrophysical Journal, namely, Wong (1974).

Basic Elements of the Toroidal Coordinate System

Given the meridional-plane coordinate location of a toroidal-coordinate system's axisymmetric anchor ring, <math>~(\varpi,z) = (a,Z_0)</math>, the relationship between toroidal coordinates and Cartesian coordinates is,

<math>~x</math>	<math>~=</math>	<math>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
<math>~y</math>	<math>~=</math>	<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
<math>~z - Z_0</math>	<math>~=</math>	<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>

This set of coordinate relations appear as equations 2.1 - 2.3 in Wong (1973). They may also be found, for example, on p. 1301 within eq. (10.3.75) of [MF53]; in §14.19 of NIST's Digital Library of Mathematical Functions; or even within Wikipedia. (In most cases the implicit assumption is that <math>~Z_0 = 0</math>.)

Mapping the other direction [see equations 2.13 - 2.15 of Wong (1973) ], we have,

<math>~\eta</math>	<math>~=</math>	<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
<math>~\cos\theta</math>	<math>~=</math>	<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
<math>~\tan\psi</math>	<math>~=</math>	<math>~\frac{y}{x} \, ,</math>

where,

<math>~r_1^2 </math>	<math>~\equiv</math>	<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math>
<math>~r_2^2 </math>	<math>~\equiv</math>	<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math>

and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.

According to p. 1301, eq. (10.3.75) of [MF53] — or, for example, as found in Wikipedia — the differential volume element is,

<math>~h_\eta h_\theta h_\psi d\eta d\theta d\psi</math>

<math>~\biggl[ \frac{a^3 \sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math>

Selected Toroidal Function Relationships

Here, we draw from the set of toroidal function relationships that have been identified as "key equations" in our accompanying Equations appendix.

① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩

①

Beginning with the identified "Key Equation",

	<math>~Q_{n-1 / 2}^m (\lambda)</math>	<math>~=</math>	<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2}~ \Gamma(n-m+1 / 2)} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, , </math>
Gil, Segura, & Temme (2000): eq. (8)

where:

<math>~\lambda \equiv x/\sqrt{x^2-1}</math>

we'll identify <math>~x</math> with <math>~\cosh\eta</math> — in which case we have <math>~\lambda = \coth\eta</math> — and switch the index notations, <math>~n \leftrightarrow m</math>. This gives,

<math>~(-1)^m \frac{\pi^{3/2}}{\sqrt{2} ~\Gamma(m-n+\frac{1}{2})} (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, . </math>

Drawing upon the Euler reflection formula for gamma functions, namely,

<math>~ \Gamma(z) ~\Gamma(1-z) </math>

<math>~\biggl|</math>

for example, if
<math>~z \rightarrow (m-n + \tfrac{1}{2})</math>

<math>~\Rightarrow ~~~\Gamma(m-n+\tfrac{1}{2})~\Gamma(n-m+\tfrac{1}{2})</math>	<math>~=</math>	<math>~\pi \biggl\{\sin\biggl[ \frac{\pi}{2} + \pi(m-n) \biggr] \biggr\}^{-1}</math>
	<math>~=</math>	<math>~\pi (-1)^{m-n} </math>

DLMF §5.5(ii)

<math>~\biggl|</math>

Valid for:

<math>~z \ne0, \pm 1, \pm 2, </math> …

<math>~\biggl|</math>

where it is understood that <math>~m</math> and <math>~n</math> are each either zero or a positive integer, this toroidal-function relation can be rewritten as,

<math>~Q_{m-1 / 2}^n (\coth\eta)</math>	<math>~=</math>	<math>~(-1)^m~ \frac{\pi^{3/2}}{\sqrt{2} } \biggl[ \frac{ \Gamma(n - m +\frac{1}{2}) }{ \pi (-1)^{m-n} } \biggr] (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) </math>
	<math>~=</math>	<math>~ (-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{1}{2} )(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, . </math>

②

Again, beginning with the identified "Key Equation",

	<math>~Q_{n-1 / 2}^m (\lambda)</math>	<math>~=</math>	<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2}~ \Gamma(n-m+1 / 2)} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, , </math>
Gil, Segura, & Temme (2000): eq. (8)

where:

<math>~\lambda \equiv x/\sqrt{x^2-1}</math>

this time, without switching index notations, we'll identify <math>~x</math> with <math>~\coth\eta</math> — in which case we have <math>~\lambda = \cosh\eta</math>. This gives,

<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} ~\Gamma(n-m+\frac{1}{2})} \biggl( \frac{1}{\sinh\eta} \biggr)^{1 / 2} P_{m-1 / 2}^n(\coth\eta) \, . </math>

Drawing upon the same Euler reflection formula for gamma functions, as quoted above, this toroidal function relation can be rewritten as,

<math>~Q_{n-1 / 2}^m (\cosh\eta)</math>	<math>~=</math>	<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} } \biggl[ \frac{\Gamma(m-n+\frac{1}{2})}{\pi (-1)^{m-n}} \biggr] \biggl( \frac{1}{\sinh\eta} \biggr)^{1 / 2} P_{m-1 / 2}^n(\coth\eta) </math>
	<math>~=</math>	<math>~(-1)^m~ \sqrt{\frac{\pi}{2}} \biggl[ \frac{\Gamma(m-n+\frac{1}{2})}{\sqrt{\sinh\eta}} \biggr] P_{m-1 / 2}^n(\coth\eta) \, . </math>

Finally, associating <math>~z</math> with <math>~ \coth\eta</math> in the "Key Equation" relation,

<math>~\biggl[ \frac{\Gamma(m+n+\frac{1}{2})}{\Gamma(m-n+\frac{1}{2})} \biggr]P^{-n}_{m-\frac{1}{2}}(z) \, ,</math>

we have,

<math>~(-1)^m~ \sqrt{\frac{\pi}{2}} \biggl[ \frac{\Gamma(m+n+\frac{1}{2})}{\sqrt{\sinh\eta}} \biggr] P_{m-1 / 2}^{-n}(\coth\eta) \, . </math>

Green's Function Expression

As presented by Wong (1973)

Referencing [MF53], Wong (1973) states that, in toroidal coordinates, the Green's function is,

<math>~\frac{1}{\|~\vec{x} - {\vec{x}}^{~'} ~\|} </math>	<math>~=</math>	<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{m,n=0} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math>
		<math>~ \times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math>
Wong (1973), p. 293, Eq. (2.53) [see also: J. W. Bates (1997), p. 3685, Eq. (31)]

where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>. This Green's function expression can indeed be found as eq. (10.3.81) on p. 1304 of [MF53], but it should be noted that the MF53 expression contains two (presumably type-setting) errors: First, the factor, <math>~(-1)^m</math>, appears as <math>~(-i)^m</math> in MF53; and, second, in the term that is composed of a ratio of gamma functions, the denominator appears in MF53 as <math>~\Gamma(n - m + \tfrac{1}{2})</math>, whereas it should be <math>~\Gamma(n + m + \tfrac{1}{2})</math>, as presented here.

Rearranging Terms

Let's focus on the situation when <math>~\eta^' > \eta</math>, and begin rearranging or substituting terms.

<math>~\frac{1}{\|~\vec{x} - {\vec{x}}^{~'} ~\|} </math>	<math>~=</math>	<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math>
		<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} ~P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') </math>
	<math>~=</math>	<math>~ \frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math>
		<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~ \sqrt{ \frac{\pi}{2} }~\Gamma(n-m+\tfrac{1}{2}) \sqrt{\sinh\eta}~P^m_{n-1 / 2}(\cosh\eta) \biggl\}\biggr\{ ~\sqrt{ \frac{2}{\pi} }~\frac{\sqrt{\sinh\eta^'}}{\Gamma(n + m + \tfrac{1}{2})} Q^m_{n-1 / 2}(\cosh\eta^') \biggr\} </math>

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Contents

Using Toroidal Coordinates to Determine the Gravitational Potential

Basic Elements of the Toroidal Coordinate System

Selected Toroidal Function Relationships

Green's Function Expression

As presented by Wong (1973)

Rearranging Terms

See Also

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